Abstract: Modern approaches to data analysis trends towards more and more exact, or qualitative descriptions of the geometric shape of data sets as a method of providing deeper insights into the processes generating the data. One fundamental approach that has gotten a lot of traction in recent years is Topological Data Analysis, and a fundamental technique for extracting and understanding shapes of data sets comes with persistent homology. In this talk, we will introduce homology, demonstrate what it measures, and how to adapt the topological techniques to a data analysis setting. This produces persistent homology, which generates a statistical summary of the topological and qualitative shape features of the data set.